Let $l$ be a rational prime, let $K/\mathbb{Q}_l$ be a finite [field extension](/page/Field%20Extension), and let

\begin{align*} \rho_1,\rho_2: G_{\mathbb{Q}}\longrightarrow GL_n(K) \end{align*}

be continuous semisimple $K$-linear representations of the absolute [Galois group](/page/Galois%20Group) $G_{\mathbb{Q}}=\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Suppose that each $\rho_i$ is unramified outside a finite set of rational primes and that there exists a finite set of rational primes $T$ such that, for every rational prime $p\notin T$ at which both representations are unramified,

\begin{align*} \operatorname{tr}\rho_1(\operatorname{Frob}_p)=\operatorname{tr}\rho_2(\operatorname{Frob}_p), \end{align*}

where $\operatorname{Frob}_p$ denotes an arithmetic Frobenius conjugacy class at $p$. Then $\rho_1$ and $\rho_2$ are isomorphic as semisimple $K$-representations of $G_{\mathbb{Q}}$.